Operator Splitting and Local Time-Stepping Methods for Transport Problems in Fractured Porous Media

Abstract

This paper is concerned with efficient numerical methods for the advection-diffusion equation in a heterogeneous porous medium containing fractures. A dimensionally reduced fracture model is considered, in which the fracture is represented as an interface between subdomains and is assumed to have larger permeability than the surrounding area. We develop three global-in-time domain decomposition methods coupled with operator splitting for the reduced fracture model, where the advection and the diffusion are treated separately by different numerical schemes and with different time steps. Importantly, smaller time steps can be used in the fracture-interface than in the subdomains. The first two methods are based on the physical transmission conditions, while the third one is based on the optimized Schwarz waveform relaxation approach with Ventcel-Robin transmission conditions. A discrete space-time interface system is formulated for each method and is solved iteratively and globally in time. Numerical results for two-dimensional problems with various Péclet numbers and different types of fracture are presented to illustrate and compare the convergence and accuracy in time of the proposed methods with local time stepping.